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Lauda & Pfeiffer on Open-Closed Topological Strings

Posted by Urs Schreiber

A closed topological 2D field theory is something which associates a vector space V of states to the circle and a linear operator V⊗n→OΣV⊗m to worldsheets Σ of closed strings with n incoming and m outgoing boundary components. More technically, this is a functor 2Cob→TVect from the category of 2-dimensional (closed) cobordisms to vector spaces.

It is very well known that such a functor is the same as a commutative Frobenius algebra. The pair-of-pants diagram with 2-incoming and one outgoing component corresponds to the product in the algebra, the co-pair-of-pants with one incoming and two outgoing components corresponds to the coproduct. The fact that only the topology of the worldsheet Σ is of relevance gives the associativity, co-associativity and the Frobenius-relation of the algebra.

More precisely, the statemement is that the category of functors of the above sort is equivalent to the category of commutative Frobenius algebras.
Apparently this has been rigorously established in

There is an obvious desire to generalize this to a theory of closed and open topological strings. Observations on the structure of such open-closed topological 2D field theories have been given by Moore and Segal. Apparently the best reference on this is still

There it is noted that open-closed 2D TFTs are still characterized by a commutative Frobenius algebra, describing the closed string sector, but now there is in addition also a not-necessarily commutative one which describes the open string sector. Furthermore there is an algebra homomorphism between the closed string sector and the center of the open string sector.

Lauda & Pfeiffer claim to fill in some gaps in making this sort of folk lore statement precise and rigorously proven.

Moore and Segal describe open-closed 2D TFT in terms of generators and relations. This means they list a set of elementary worldsheets, like the cylinder, the pair of pants and the disc, and then define all acceptable worldsheets as those obtainable from composing these in all different ways while identifying those that are related by certain relations. These relations are induced by worldsheet diffeomorphisms. Hence they are necessary if the generators and relations-description is to capture all of 2D TFT. Lauda and Pfeiffer observe that it has not been previously established rigorously that the set of relations considered is also sufficient for this purpose - that every diffeomorphism between open-closed worldsheets can be obtained from combining the listed relations between elementary worldsheets. Proving this is their first result.

With this result in hand, one can define the category 2Cobext of open-closed 2D cobordisms. An open-closed 2D TFT is then a functor 2Cobext→TC, where C is some symmetric monoidal category, like Vect for instance. One would again like to know what algebraic structure such functors encode. Lauda & Pfeiffer’s second result is that the category of these functors is equivalent to the category of what they call knowledgeable Frobenius algebras.

A knowledgeable Frobenius algebra is a commutaive Frobenius algebra describing the closed sector, together with a ‘symmetric’ Frobenius algebra describing the open sector, equipped with morphism between these two such that some relations hold. These morphisms capture the action of worldsheets where a closed string splits into an open string (the ‘zipper’) and the reverse situation where an open string merges its ends to a closed string (the ‘co-zipper’).

As far as I understand, this result is new in that it really goes a little beyond what Moore and Segal state in theorem 1 on their slide 29 in the above lecture notes.

One of the main tools used in the paper is a generalization of Morse theory to ‘manifolds with corners’.

The elementary worldsheet segments - the generators - of a closed topological string can be obtained as the neighborhoods of critical points of any one Morse function on the worldsheet. Different choices of Morse functions correspond to moves between different choices of constructing a worldsheet from elementary worldsheets.

Open-closed worldsheets have boundaries and ‘corners’ where an incoming open string ends on a D-brane. There is an old generalization of Morse theory to such a situation invented by Braess in 1974. Using this, Lauda and Pfeiffer can deduce the generating worldsheets of open-closed strings. It takes some time to write all this out in detail, but the result is just what one expects.

The well-known relations hold between compositions of these generators, obtainable by applying diffeomorphisms to them. The biggest chunk of the paper is concerned with defining a certain ‘normal form’ of composites of generators for open-closed worldsheets and proving that using these relations every worldsheet can be brought into its normal form. This proves that the relations one has are sufficient in that they are all that are needed to emulate every possible diffeomorphism in terms of moves between composites of generators.

With the category 2Cobext thus captured in terms of generators and relations, it is pretty straightforward to define open-closed TQFTs as functors from these to some symmetric monoidal category C and check that such a functor defines a knowledgeable Frobenius algebra (object in C). The generalization to the presence of more than one D-brane is also rather immediate.

Posted at November 13, 2005 2:31 PM UTC

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Re: Lauda & Pfeiffer on Open-Closed Topological Strings

Urs writes:

As far as I understand, this result is new in that it really goes a little beyond what Moore and Segal state in theorem 1 on their slide 29 in the above lecture notes.

Actually, the main new thing that Lauda and Pfeiffer do is prove their result: Moore and Segal just state theirs. So, Lauda and Pfeiffer deserve credit for turning a plausible conjecture into a theorem.
It’s easy to guess a lot of relations that these open-closed 2d cobordisms satisfy, and after a while of playing around one can feel sure that one has gotten a sufficient set. However, proving this takes work - and that’s what Lauda and Pfeiffer do.

I’m not sure, but it’s also possible that Moore and Segal forgot one or two relations which Lauda and Pfeiffer found. One virtue of proving theorems is that one catches little mistakes like this….

It also takes some interesting category-theoretic machinery to state precisely what one means by saying that one has gotten a sufficient set of generators and relations for a symmetric monoidal category. A slight variant of Lawvere’s “algebraic theories” (namely, a so-called “PROP”) does the trick.

Lauda and Pfeiffer are working on some other papers which go further….

Re: Lauda & Pfeiffer on Open-Closed Topological Strings

I think the most interesting results along these lines right now is Costello’s.

First one generalizes the usual definition of a CFT to get something on the level of chain complexes. In particular, one makes a (dg) category whose objects are labelled by the nonnegative integers (as usual), but whose morphisms are the space of chain complexes on the moduli space of Riemann surfaces with the relevant set of boundaries.

A closed Toplogical CFT (TCFT) is then a differential graded symmetric monoidal functor from this category to the categories of chain complexes (ie, dg-Vect, I think).

Now, define a CY category to be a category of dimension d to be a k-linear category with a trace from Hom(A,A)→k[d].

Throw in a bunch of A∞s and you get the following very cool theorem:

The category of open TCFTs (of dimension d – I didn’t define this) with a fixed set of D-branes is homotopy equivalent to the category of A∞ categories if dimension d with objects being the D-branes.

One can also show that the closed string states are exactly given by the Hochschild homology of the A∞ category.

Costello’s strings

Kevin Costello had written about this on the SCT a while ago, but I never took the time to try to really absorb it.

In particular, I wasn’t quite sure why this statement

There’s a category, whose objects are the integers, and whose morphisms are the singular chains on the moduli spaces of Riemann surfaces with n incoming and m outgoing boundaries. A functor from this to chain complexes, which is compatible with differentials, should be a closed topological string theory.

Read the post (String) Physics from (Higher) AlgebraWeblog: The String Coffee TableExcerpt: One further refinement of the tale called 'Strings from categorified quantum mechanics.'Tracked: January 9, 2006 8:56 PM